The element $q$ of a ring $R$ is called quasi-idempotent if $q^2=uq$ for some central unit $u$ $R$, or equivalently $q=ue$, where and $e$ an idempotent $R$. In this paper, we define that the almost quasi-clean each sum regular element. Several properties almost-quasi clean rings are investigated. We prove every quasi-continuous nonsingular quasi-clean. Finally, it determined conditions under wh...

We presented a simple and direct way to construct unipotent unit clean but not nil-clean element in ring. New examples of unipotent/unit-regular elements that are given. also study the product two idempotents/unit-regulars which unit-regular. The studies exemplified subrings M_2 (Z).

In this paper, we study the transfer of notion $m$-clean ring (i.e., a in which every element is sum unit and an $m$-potent element) to amalgamated bi-amalgamated rings. We also find many sufficient necessary conditions for m-clean property Received: March 7, 2023Accepted: May 31, 2023

1-Carboxymethyl-3-methylimidazolium chloride {[cmmim]Cl} is utilized as a highly efficient and reusable catalyst for the synthesis of bis(indolyl)methanes by the condensation reaction of indole with arylaldehydes under solvent-free conditions at room temperature. In summary, the presented method have several advantages including simplicity, low cost, generality, efficiency, clean reaction profi...

1-carboxymethyl-3-methylimidazolium chloride {[cmmim]cl} is utilized as a highly efficient and reusable catalyst for the synthesis of bis(indolyl)methanes by the condensation reaction of indole with arylaldehydes under solvent-free conditions at room temperature. in summary, the presented method have several advantages including simplicity, low cost, generality, efficiency, clean reaction profi...

THE FEASIBILITY OF TRANSFERRING CLEAN TECHNOLOGY FROM THE UNITED STATES TO CHINA: A CASE STUDY FROM THE PAPER INDUSTRY

reduces the congestion and makes the proceedings easier. Technique.?First clean the part with rectified spirit. Starting at the tip of the finger (figure 1) a strong thread (twine would suffice) is wound round the fingers, the strands touching each other without overlapping, till the strands reach in the vicinity of the ring. The free end of the thread is now manipulated under or preferably thr...

The main results: A ring R is CN if and only if for any x ∈ N(R) and y ∈ R, ((1+x)y)n+k = (1+x)n+kyn+k, where n is a fixed positive integer and k = 0, 1, 2; (2) Let R be a CN ring and n ≥ 1. If for any x, y ∈ R\N(R), (xy)n+k = xn+kyn+k, where k = 0, 1, 2, then R is commutative; (3) Let R be a ring and n ≥ 1. If for any x ∈ R\N(R) and y ∈ R, (xy)k = xkyk, k = n, n + 1, n + 2, then R is commutati...